Question

Prove that the trajectory of a projectile is parabolic, having the form $\mathit{y}\mathbf{=}\mathit{a}\mathit{x}\mathbf{+}\mathit{b}{\mathit{x}}^{\mathbf{2}}$. To obtain this expression, solve the equation $\mathit{x}\mathbf{=}{\mathit{v}}_{\mathbf{o}\mathbf{x}}\mathit{t}$for and substitute it into the expression for $\mathit{y}\mathbf{=}{\mathit{v}}_{\mathbf{o}\mathbf{y}}{\mathit{t}}_{\mathbf{1}}\mathbf{(}\mathbf{1}\mathbf{/}\mathbf{2}\mathbf{)}\mathit{g}{\mathit{t}}^{\mathbf{2}}$(These equations describe the xand y positions of a projectile that starts at the origin.) You should obtain an equation of the form $\mathit{y}\mathbf{=}\mathit{a}\mathit{x}\mathbf{+}\mathit{b}{\mathit{x}}^{\mathbf{2}}$where a and bare constants.

Short answer

The trajectory of the displacement is proved to be a parabolic path.

When gravity first exerts force on an item, its initial velocity indicates how fast it travels. The final velocity, on the other hand, is a vector number that measures a moving body's speed and direction after it has reached its maximum acceleration.

Step-by-step solution

Definition of initial velocity

The velocity can be said as.$v_i$ The velocity is making some angle with the x axis.

In the x frame the acceleration are zero, hence the velocity in the x frame will be constant.

Proving parabolic trajectory of displacement

The initial velocity and the final velocity and the average velocity will be the same.

$$\begin{gathered} \overline{V}=\frac{X}{t} \\ t=\frac{X}{V_i\cos \theta }.......1 \end{gathered}$$

Now let’s calculate the displacement.

Hence the trajectory of the displacement is proved to be a parabolic path.